Of course not. Judo is a martial art. BJJ, Sambo, TKD, Shotokan Karate... all martial arts. When you mash them together and use them in a cage under the unified ruleset of MMA, they remain martial arts, but the sum becomes a sport.
Or a Martial Art. Kajukenbo, for example. Or Hapkido. Or even Taekwondo, since although it was primarily derived from Shotokan, there are also other influences.
Genre typically refers to a sub category of an art form: music, literature, film, painting, sculpture. If you want to push the "art" in martial art, I get where you're going, but it's not really the same thing.
That said, it's not worth arguing. I understand how you're using the term, and that's what really matters. Given your use of the term, I still disagree that MMA is a genre of martial art. If anything, it's a sport that incorporates many genres of MA according to various formulas... recipes if you will.
Well, it wasn't my use. I just quoted it and didn't think it was worth picking the nit. It's close enough that I can see an argument for it's use.